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Lecture 2 Propositional Logic and Proof Procedure
CS6133Software Specification and
Verification
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Propositional Logic as Formal Language
A logic consists of Syntax: define well-formed formula Semantics: define meaning of formula
interpretation of logical connectives
satisfaction relation
semantic entailment Proof procedure (also called proof theory)
Soundness
Completeness
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Propositional Logic Syntax The syntax elements
Two constant symbols: true and false Propositions: A, B Logic connectives
Brackets
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Propositional Logic Syntax A well-formed formula (WFF) of propositional logic is constructed as below
Every proposition p is a WFF If P is a WFF, then so is (P) If P is a WFF, then so is P If P and Q are WFFs, then so is P Q If P and Q are WFFs, then so is P Q If P and Q are WFFs, then so is P Q
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Propositional Logic Semantics Semantics mean “meaning” and relate two worlds: provide an interpretation (mapping) of expressions in one world in terms of values in another world
Semantics are often a function from expressions in one world to expressions in another world
The range of the semantic function for propositional logic is the set of truth values
Tr = {TRUE, FALSE}
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Truth Table Truth assignmentA truth assignment is a mapping of the variables within
a
formula into the value TRUE or FALSE
Truth tables are used to describe the functions of logic connectives on the truth values
Truth tables determine the truth value of logic formulas
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Propositional Logic Semantics Satisfiable A formula is satisfiable if there exists some truth
assignment under which the formula has truth value TRUE
Valid A formula is valid or a tautology, if it has truth value
TRUEunder all possible truth assignments
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Satisfaction and Entailment Satisfaction relation A model M satisfies the formula P is called a
satisfaction
relation
M P
Entailment relation From the premises P1 , P2 , P3 , … , we may conclude Q,
where
P1 , P2 , P3 , … and Q are all well-formed propositional
logic
formulas P1 , P2 , P3 Q
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Decidability A logic is decidable if there is an algorithm to determine if any formula of the logic is a tautology (is a theorem, is valid)
Propositional logic is decidable because we can always construct the truth table for the propositional formula
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Propositional Logic Proof Procedure
A proof procedure is a set of rules we use to transform premises and conclusions into new premises and conclusions
A goal is a formula that we want to prove is a tautology
A proof is a sequence of proof rules that when chained together relate the premise of the goal to the conclusion of the goal
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Truth Table vs. Proof Procedure Determine if a formula is a tautology by using truth tables: determine the value of the formula for every possible combination of values for its proposition letters
Constructing truth table would be very tedious since the size of the truth table grows exponentially: it is NP-complete
Proof procedures for propositional logic are alternate means to determine tautologies
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Example Proof Procedures Hilbert Systems: axiom systems
Natural Deduction
Binary Decision Diagrams
Sequent Calculus
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Hilbert System A Hilbert system consists of
Axioms: a set of valid formulas Inference rules
Inference Rules Determine tautology or unsatisfiability Manipulate formulas as formal strings of symbols But do not make use of the meanings of formulas
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Proof in Hilbert System Proof is a finite sequence X1, X2, … ,Xn of formulas such that each term is either an axiom or follows from earlier terms by one of the rules of inference
Write proofs as a list of formulas, each on its own line, and refer to the line of a proof in the justification for steps
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Hilbert System Hilbert system is sound
If start with axioms (which are valid) Then each subsequent formula derived with
inference rules is also valid
Hilbert system is complete If start with axioms (which are valid) Then it can derive all formulas with are valid
Hilbert system is consistent If start with axioms (which are valid) Then it is impossible to prove both P and P
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An Axiomatic System for Propositional Logic
Three axioms A (B A) (A (B C)) ((A B) (A C)) ( A B) (B A)
One rule of inference
From A and A B, B can be derived, where A and B are any well-formed formulas
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Exercise Show (X Y) (X X)
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Natural Deduction A collection of proof rules, each of which allows us to infer formulas from other formulas, eventually to get from a set of premised to a conclusion
A form of forward proof Starting from the premises Use the inference rules to deduce new formulas that
logically follow from the premises Continue this process until we have deduced the
conclusion
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Natural Deduction Rules Rules for conjunction
Rules for double negation
Rules for eliminating implication: modus ponens
Rule implies introduction
Rules for disjunction
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Natural Deduction RulesRules for conjunction
p q _______ i p q
p q p q _______ e1 _______ e2
p q
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Natural Deduction RulesRules for double negation
p p _____ e _____ i p p
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Natural Deduction RulesRules for eliminating implication
p p q p q q _______ e ________ e
q p
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Natural Deduction RulesRule implies introduction
p .
. .
q _______ i
p q
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Natural Deduction RulesRules for disjunction
p q _______ i1 _______ i2 p q p q
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